Fast Geodesic Active Fields for Image Registration based on Splitting and Augmented Lagrangian Approaches

In this paper we present an efficient numerical scheme for the recently introduced Geodesic Active Fields (GAF) framework for geometric image registration. This framework considers the registration task as a weighted minimal surface problem. Hence, data-term and regularization term are combined trough multiplication in a single, parametrization invariant and geometric cost functional. The multiplicative coupling provides an intrinsic spatially varying and data-dependent tuning of the regularization strength, while the parametrization invariance allows working with images of non-flat geometry, generally defined on any smoothly parametrizable manifold. The directly resulting minimizing energy flow, however, has poor numerical properties. Here, we provide an efficient numerical scheme that uses a splitting approach: data and regularity term are optimized over two distinct deformation fields that are constrained to be equal via an augmented Lagrangian approach. Our approach is more flexible than standard Gaussian regularization, since one can interpolate freely between isotropic Gaussian and anisotropic TV-like smoothing. In this work, we compare the Geodesic Active Fields method against the popular and fast state-of-the-art Demons method. Overall, we can show the advantages of the proposed FastGAF method. It compares favorably against Demons, both in terms of registration speed and quality.